The magic of mathematical intuition

Why the math you do understand feels stupidly easy

Take a billion. Then take away one. How much is left?

You don’t really need to think. You can see the answer in your head: 999,999,999. The answer is easier to picture than it is to pronounce.

It seems obvious, and yet it wasn’t always like that. To someone living in ancient Rome, for example, it wouldn’t have been obvious at all.

In classical Latin, the word billion didn’t exist (neither did million). To communicate the idea, the easiest thing would have been to call it the product of “a thousand times a thousand times a thousand.” A Roman during the time of Julius Caesar should have been able to understand that, even if it might have given them a bit of a headache. But if you had told them that you were capable of taking this number, subtracting one from it, and picture the answer immediately in your head, they wouldn’t have been able to follow.

They would have taken you for some kind of math whiz.

You’d be hard pressed to write 999,999,999 in Roman numerals. If Roman numerals are the only numbering system you know, 999,999,999 is much more than a big number you don’t run into every day. It’s a number that you can’t even “look” at. It’s so terrifying that it makes your head spin. The idea that someone could instantly “see” it clearly and without any effort is absurd.

But there’s nothing extreme about the ancient Romans. Their understanding of numbers was really quite advanced. The traditional way of counting among certain aboriginal Australian people is based on parts of the body. You count from 1 to 5 on the fingers, then move up the arm: 6 is the wrist, 7 the forearm, 8 the elbow, 9 the biceps. When you get to 10 (the shoulder), you keep going up the body—12 is the earlobe. Yet if each number needs a corresponding body part, how do you get to a billion?

In the Amazon, Yanomami languages have an even more restricted numeral system: there’s a word for “one” and another for “two,” but there’s no word for “three,” just a catchall word that basically means “a lot.”

For someone who sees the world in this way, discovering that there’s a clear distinction between 25 and 26 that can be perceived in a split second must come as something of a revelation, comparable to what math students experience when they learn that there are many different sizes of infinity that can be precisely described.

A Complete Sham?

An inhabitant of ancient Rome would be able to grasp immediately the difference between XXV and XXVI. But your agility with big numbers would lead them to believe that you’re a math whiz. That idea makes you smile, because you know for certain that you’re no math whiz.

But are you sure about that?

If you think a math whiz is some kind of mutant with supernatural powers, if you think that they have some kind of computer in their head that lets them do calculations super quickly using the same methods that you know, then you’re wrong.

In the end, math whizzes are kind of like Santa Claus: they don’t really exist. When you think you’ve seen Santa, it’s never really Santa, just someone dressed up like him. When you think you’ve seen a magician, it’s never really a magician, it’s always an illusionist, someone who knows tricks that can create the illusion that they have magical powers.

And when you think you see a math whiz, it’s never really a math whiz, it’s always just someone who has a way of seeing numbers that turns calculations that you find complex and scary into something easy and even obvious.

The truth is that we’re all basically bad at mental calculation, except when we have an intuitive way of radically simplifying the calculation and “seeing” the result.

The decimal system based on Hindu-Arabic numerals is a “trick” that lets us see certain results as obvious. The main difference between a math whiz and you is that their bag of tricks is bigger than yours and they’re more used to playing with them.

Real Understanding

The decimal system of writing numbers seems so obvious to you that you can’t even remember learning it. It’s just like using a spoon. You use it without really thinking about it, like it’s an extension of your own body. When you see 999,999,999, you think you’re seeing the number directly, without realizing that you’re seeing it with the help of a tool.

Decimal writing is a purely human invention. More than simply a system of writing, it’s a door into a state of consciousness where whole numbers, however big they may be, become concrete and precise objects. At the same time, the infinitude of whole numbers becomes commonplace.

Something previously unimaginable suddenly becomes commonplace: this is exactly the type of effect mathematics produces in your brain. It’s a marvelous sensation, a great delight.

When you were a child, you were proud to be able to count to 10, then 20, then 100. It gave you bragging rights at recess. In order to brag some more, you would have wanted to know the biggest number.

To tell the truth, your awareness of numbers wasn’t that far off from those people who can count to 2 or 5 and are firmly convinced that the next number, the number many, is the biggest number.

One day, you realized that no number was the biggest. Even if you might have arrived at this conclusion some other way, decimal writing gave you a shortcut. You know that every number is followed by another. You know how to see the succession of numbers like a counter that turns, and you know that this counter can turn indefinitely. There’s no limit, there’s no special number after which the counter stops working.

Yet for 99 percent of human history, no one had been able to picture a number counter turning in their head.

The number counter turning in your head is the collective work of great mathematicians who, from prehistory until the Middle Ages, fashioned the image of numbers that we share today.

This image isn’t natural. It wasn’t inscribed in your body the day you were born. It’s partially arbitrary: we might have chosen another system for writing numbers, and you would see them differently.

More than four thousand years ago the Babylonians invented a sexagesimal system: they wrote their numbers in base 60 rather than base 10. Babylonian mathematicians were the most advanced of their time. Your mental image of hours, minutes, and seconds remains profoundly influenced by their vision of numbers.

What is natural, however, is your capacity to assimilate abstract mathematics and to really understand them, to modify your brain so that this math really becomes part of you.

You believe you can see the number 999,999,999. What you’re really doing is deciphering a complex and abstract mathematical notation. You decipher it instantly, fluently, without even realizing it. Whole numbers may not be your mother tongue, but you’ve become bilingual.

Successful math becomes so intuitive that it no longer feels like math. If the example seems stupid to you, it’s precisely because you understand it at the deepest level.

Real Magic Doesn’t Exist

At the start of their careers, young mathematicians often feel like imposters.

It’s a feeling I know well, and in my case it seemed entirely justified. The results contained in my PhD thesis were so obvious that it was almost like a trick. My theorems were always simple, and their proofs never contained any real difficulties.

Everyone around me seemed to be better at math. They were working on profound stuff that was way out of my league. They were writing papers that were extremely difficult to read, with proofs that seemed incredibly complex and technical. If I managed to understand a few of them, it’s only because they happened to be easier than usual.

I wanted to know how to do real math, difficult math. But all that I was able to learn was the easy math, the math for dummies.

It seems silly to say this, but it really took me years to realize it was only an optical illusion. The horizon was shifting with me. It was always staying at my level.

Real magic doesn’t exist. When you learn a magic trick, it ceases being magical. That may be sad, but you’d better get used to it.

If you find that the math you do understand is too easy, it’s not because it’s easy, it’s because you understand it.

This post is an excerpt (Chapter 4) from Mathematica, A Secret World of Intuition and Curiosity, Yale University Press (2024).